Exact Optimal Designs of Experiments for Factorial Models via Mixed-Integer Semidefinite Programming

Abstract

The systematic design of exact optimal designs of experiments is typically challenging, as it results in nonconvex optimization problems. The literature on the computation of model-based exact optimal designs of experiments via mathematical programming, when the covariates are categorical variables, is still scarce. We propose mixed-integer semidefinite programming formulations, to find exact D-, A- and I-optimal designs for linear models, and locally optimal designs for nonlinear models when the design domain is a finite set of points. The strategy requires: (i) the generation of a set of candidate treatments; (ii) the formulation of the optimal design problem as a mixed-integer semidefinite program; and (iii) its solution, employing appropriate solvers. For comparison, we use semidefinite programming-based formulations to find equivalent approximate optimal designs. We demonstrate the application of the algorithm with various models, considering both unconstrained and constrained setups. Equivalent approximate optimal designs are used for comparison.

Model robust designs for dose-response models

“…Contrastingly, convex programming algorithms have not been hitherto applied in the construction of model-robust designs. Their application has gained prominence and solidified as a well-established methodology, finding utility in constructing (i) locally optimal designs (Duarte et al, 2018b;Papp, 2012;Sagnol, 2011;Vandenberghe and Boyd, 1999); (ii) Bayesian optimal designs, offering protection against parametric uncertainty (Duarte and Wong, 2015); (iii) minmax optimal designs (Duarte et al, 2018a); and (iv) exact optimal designs (Duarte, 2023;Sagnol and Harman, 2015). Our analysis reveals an unexplored space for applying convex programming-based methodologies in the domain of model-robust designs.…”